These weeks are called שובבים. That is an acronym for the parshiyos שמות וארא בא בשלח יתרו משפטים. This is based on the pasuk שובו בנים שובבים - Return wayward sons. During these weeks we are supposed to do extra teshuva. Heavy duty. Sort of like a winter-dike Elul. The source for this is not the gemara but the later mekubalim. Those who can appreciate kabbala in its depth can appreciate שובבים. I am not a mekubal. To be a true mekubal you need to enjoy chicken and a hot cup of tea a lot less than I do. But I try to dabble in sifrei chassidus which elevate me a little out of my coarse physicality as I suggest everyone does.
I once went to a mekubal [Rav Gamliel Rabinowitz Shlita] and he gave me a siddur he published of special mystical tefillos to say during this period. Some people fast every day [until night] or at least Monday and Thursday. The Klausenberger Rebbe told his wife at this time that for the next six weeks she is off duty [meaning that he is not going to eat]. I knew a tzaddik who fasted every day until mincha - but he davened mincha right before shkiya.
The area in which we place a special emphasis is in the area of arayos, histaklus and all around kedusha. These are the parshiyos of Mitzraim which was called ערות הארץ - The nakedness of the land. We try to put some clothing on this nakedness....
This was a shiur on this topic geared for married people.
If people want more on this topic email me and I will try to write more. I also recommend you purchase the sefer on שובבים written based on the sichos of Rav Tzvi Meir Zilberberg who is an angel living amongst men.
For gematria fans -
If we do teshuva we reach שלמות which eliminates the power of the shor [Esav] and chamor [Yishmael] which will ultimately bring Moshiach.
They are all the exact same gematria -
שובו בנים שובבים = שלמות = שור וחמור = ביאת המשיח 776